The key to finding the secure key rates is to find the minimum value of \(D\left(\mathcal {G}\left(\rho _{A B}\right) \Vert  \mathcal {Z}\left[\mathcal {G}\left(\rho _{A B}\right)\right]\right)\) , since \(p_{\mathrm {pass}} \delta _{\mathrm {EC}}\)  is a fixed quantity. The associated optimization problem is [1]}
\(\begin{split}\operatorname{minimize} &D\left(\mathcal {G}\left(\rho _{A B}\right) \Vert  \mathcal {Z}\left[\mathcal {G}\left(\rho _{A B}\right)\right]\right) \\\text{subject to } &\\&\operatorname{Tr}\left[\rho _{A B}\left(|x\rangle \left\langle \left. x\right|_{A} \otimes \hat{q}\right)\right]=p_{x}\langle \hat{q}\rangle _{x}\right., \\&\operatorname{Tr}\left[\rho _{A B}\left(|x\rangle \left\langle \left. x\right|_{A} \otimes \hat{p}\right)\right]=p_{x}\langle \hat{p}\rangle _{x}\right., \\&\operatorname{Tr}\left[\rho _{A B}\left(|x\rangle \left\langle \left. x\right|_{A} \otimes \hat{n}\right)\right]=p_{x}\langle \hat{n}\rangle _{x}\right., \\&\operatorname{Tr}[\rho _{A B}(|x\rangle \left\langle x\right|_{A} \otimes \hat{d})]=p_{x}\langle \hat{d}\rangle _{x}, \\&\operatorname{Tr}\left[\rho _{A B}\right]=1, \\&\rho _{A B} \ge 0,\\&\operatorname{Tr}_{B}\left[\rho _{A B}\right]=\sum _{i, j=0}^{3} \sqrt{p_{i} p_{j}}\left\langle \varphi _{j} \mid \varphi _{i}\right\rangle |i\rangle \left\langle \left. j\right|_{A}\right.,\end{split}\) 
